HELPP Math word Problem please !!

Question

All the problems we have looked at in this chapter have assumed that all participants receive an equal share of what is being divided. Often, this does not occur in real life. For example, Grandma might specify in her will that John is to receive 50%, Sue is to receive 30%, and Margaret is to receive 20%.

To make things more reasonable, lets look at a simpler example. Suppose Fred and Maria are going to divide a cake using the divider-chooser method. However, Fred is only entitled to 30% of the cake, and Maria is entitled to 70% of the cake (maybe it was a $10 cake, and Fred put in $3 and Maria put in $7). Adapt the divider-choose method to allow them to divide the cake fairly. Assume (as we have throughout this chapter) that different parts of the cake may have different values to Fred and Maria, and that they don't communicate their preferences/values with each other. 

Clearly convey your method. The methods in the book use a step-by-step explanation. You might find doing the same helpful.

Your goal is to come up with a method of fair division, meaning that although the participants may not receive equal shares, they should be guaranteed their fair share. Your method needs to be designed so that each person will always be guaranteed a share that they value as being worth at least as much as they're entitled to.

Answer 1

The Method of Ratios

Step 1: Determine the total number of sub-pieces

Add the ratios of the two participants together to find the total number of pieces needed.

1. Fred: 3 parts
2. Maria: 7 parts
3. Total: 10 parts

Step 2: The Division

One person is designated as the Divider (let's say, Maria). Maria must cut the cake into 10 pieces that she considers to be of equal value.

1. Note: The pieces do not have to be the same size or shape, but in Maria’s eyes, every single piece must be worth exactly 10% of the total value.

Step 3: The Selection

The other person is designated as the Chooser (Fred). Fred examines the 10 pieces Maria has cut. He gets to choose any 3 pieces he wants.

Step 4: The Conclusion

Maria takes the remaining 7 pieces.

For a division method to be fair, both parties must believe they received at least the value they were entitled to, regardless of what the other person values.

Why it is fair for Maria (The Divider):

Maria cut the cake into 10 pieces. Because she was the Divider, she ensured that every piece was worth exactly 10% of the total cake in her eyes.

1. Fred took 3 pieces (30% value to Maria).
2. Maria was left with 7 pieces.
3. 7 pieces×10% value each=70%7 pieces×10% value each=70%.
4. Maria is guaranteed exactly 70% of the value according to her own system.

Why it is fair for Fred (The Chooser):

Fred walks up to a cake cut into 10 pieces. To him, the pieces might not be equal. Some might have more frosting (which he loves), and some might have none. However, the total value of all 10 pieces must add up to 100%.

1. If Fred values every piece equally, he picks any 3 and gets exactly 30%.
2. If Fred values the pieces differently, he will naturally pick the 3 pieces he thinks are the best.
3. Mathematically, it is impossible for the best 3 pieces to be worth less than 30% of the total. (For example, you cannot have a set of numbers that sum to 100 where the average is 10, but the largest numbers are smaller than the average).
4. Therefore, Fred is guaranteed a share worth at least 30% (and likely more, if Maria's cuts differ from his preferences).